Deck 7: Right Triangles and Vectors

Question 1

Convert 1.74 rev to radians.

Answer 1

$10.9 \mathrm{rad}$

Question 2

Express a rotation of 18.7 radians into revolutions.

Answer 2

2.98 rev

Question 3

Convert 4.71 rad to degrees (decimal).

Answer 3

$27 \overline{0}^{\circ}$

Question 4

Express the angle of $3.6 \mathrm{rad}$ in degrees, minutes, and seconds.

Question 5

What is $168.475^{\circ}$ in degrees, minutes, and seconds?

Question 6

Convert $49^{\circ} 16^{\prime} 29^{\prime \prime}$ to degrees (decimal) to 4 decimal places.

Question 7

Convert $49^{\circ} 16^{\prime} 29^{\prime \prime}$ to revolutions. Keep 3 significant digits.

Question 8

Convert $49^{\circ} 16^{\prime} 29^{\prime \prime}$ to radians. Keep 3 significant digits.

Question 9

Convert $209.6^{\circ}$ to radians.

Question 10

Convert 1.42 rad to degrees.

Question 11

Convert 0.21654 rev to degrees, minutes, and seconds.

Question 12

Blades on an in-water tidal turbine turn at 18 revolutions per minute. How fast is this in radians per second?

Question 13

The arc length, $s$ , subtended by an angle $\theta$ in a circle of radius $r$ is given by $s=r \theta$ when the angle is measured in radians. What is the arc length subtended by a $55.3^{\circ}$ angle in a circle of radius $4.20 \mathrm{~cm}$ ?

Question 14

The point $(6,8)$ is on the terminal side of angle $\theta$ . Find the distance $r$ from the origin to the point and compute $\sin \theta, \cos \theta$ , and $\tan \theta$ . Work to 3 significant digits.

Question 15

The point $(11,5)$ is on the terminal side of angle $\theta$ . Find the distance $r$ from the origin to the point and compute $\sin \theta, \cos \theta$ , and $\tan \theta$ . Work to 3 significant digits.

Question 16

Evaluate $\cos 74.6^{\circ}$ to 4 decimal places.

Question 17

Evaluate $\csc 61.7^{\circ}$ to 3 significant digits.

Question 18

Evaluate sec $1.95^{\circ}$ to 2 decimal places.

Question 19

Evaluate $\sin 74^{\circ} 43^{\prime}$ to 4 significant digits.

Question 20

Evaluate $\tan 16.5^{\circ}$ to 2 significant digits.

Question 21

Evaluate cot $186^{\circ}$ to 3 decimal places.

Question 22

Given $\theta=37.7^{\circ}$ , calculate $\sin \theta, \cos \theta$ , and $\tan \theta$ to 3 significant digits.

Question 23

Given $\theta=1.5 \mathrm{rad}$ , calculate $\sin \theta, \cos \theta$ , and $\tan \theta$ to 4 decimal places.

Question 24

Write $\sin \theta, \cos \theta$ , and $\tan \theta$ to 3 significant digits: $\theta=58^{\circ} 18^{\prime} 45^{\prime \prime}$

Question 25

Evaluate to three significant digits: $5.7 \tan 2.8^{\circ}+13.3 \sin 6.7^{\circ}$

Question 26

Evaluate to three decimal places: $12 \cos 63^{\circ}-19 \tan 24^{\circ}$

Question 27

Evaluate to four significant digits: $10.1 \csc 28.9^{\circ}+33.2 \sin 64.0^{\circ}$

Question 28

Evaluate to four decimal places: $\cot 63^{\circ}-\sec 82^{\circ}$

Question 29

If $\sin \theta=0.416$ , what is $\theta$ in decimal degrees?

Question 30

If $\sec \theta=1.12$ , what is $\theta$ in decimal degrees?

Question 31

If $\tan \theta=4.00$ , what is $\theta$ in radians?

Question 32

If $\csc \theta=15.4$ , what is $\theta$ in radians?

Question 33

If $\cot M=\frac{5}{12}$ , calculate $\sin M, \cos M$ , and $\tan M$ without using a calculator or tables. Keep your answers as fractions.

Question 34

Determine the value of $\theta$ to 3 significant digits if $\cot \theta=24.7$ .

Question 35

When the cos of an angle equals 0.3689 , find the angle (in decimal degrees) to 4 significant digits.

Question 36

Find arcsec 16.8 in degrees to 4 significant digits.

Question 37

Determine the value of $\theta$ (in decimal degrees) given that the opposite side is $2.35 \mathrm{~cm}$ and the adjacent side is $5.25 \mathrm{~cm}$ .

Question 38

Determine the value of $\theta$ (in radians) given that the opposite side is $6.5 \mathrm{~cm}$ and the hypotenuse is $8.3 \mathrm{~cm}$ .

Question 39

If $\cos A=\frac{8}{37}$ , write the sin and tan of angle $A$ without using a calculator. Leave your answer in fractional form.

Question 40

If $\tan \theta=1.428$ and the adjacent side of $\theta$ is $267 \mathrm{~cm}$ , find the length of the opposite side of $\theta$ .

Question 41

Determine all the missing angles and side lengths, given that $c=714$ and $B=56.3^{\circ}$ . Round to 3 significant digits.

Question 42

Determine all the missing angles and side lengths, given that $a=194$ and $A=0.867$ rad. Round to 3 significant digits.

Question 43

Sketch triangle $A B C$ where $a=56.8, c=88.3$ , and $C=90^{\circ}$ . Find all the missing angles and side lengths, rounding to 3 significant digits.

Question 44

Solve the right triangle $A B C$ if $A=68.2^{\circ}, B=90^{\circ}$ , and $c=23$ .

Question 45

Express $\csc 61.0^{\circ}$ as a function of the complementary angle.

Question 46

Express $\csc 34.3^{\circ}$ as a function of the complementary angle.

Question 47

Express $\sin 47^{\circ} 58^{\prime}$ as a function of the complementary angle.

Question 48

Express $\tan 62^{\circ}$ as a function of the complementary angle.

Question 49

Express cot $14^{\circ}$ as a function of the complementary angle.

Question 50

Express $\cos 16^{\circ}$ as a function of the complementary angle.

Question 51

Express sec $78^{\circ}$ as a function of the complementary angle.

Question 52

Express $\cos \pi / 4$ as a function of the complementary angle.

Question 53

Express $\cot 3 \pi / 8$ as a function of the complementary angle.

Question 54

The angle of elevation from an observer on the ground to the top of a $408 \mathrm{~m}$ high building is $38.3^{\circ}$ . How far is the observer from the bottom of the building?

Question 55

A tree casts a shadow that is $28.6 \mathrm{~m}$ long when the sun is at an angle of elevation of $57.4^{\circ}$ . How tall is the tree?

Question 56

A ship travelling due east sights a lighthouse at $\mathrm{N} 76^{\circ} 33^{\prime} \mathrm{E}$ . After 2 hours, 48 minutes the ship reaches a rock that the map says is $35 \mathrm{~km}$ due south of the lighthouse. How fast is the ship travelling, in $\mathrm{km} / \mathrm{h}$ ?

Question 57

Calculate the height of drawbridge $\mathrm{A}$ .

Question 58

The symmetric roof of a garden shed has a 5-12 pitch. Calculate the interior angles to the nearest tenth of a degree.

Question 59

Given a right triangle with hypotenuse $164 \mathrm{~cm}$ long and an adjacent side $128 \mathrm{~cm}$ long, find the angle and the length of the opposite side.

Question 60

The angle of elevation of the top of a building from a point on the ground $86 \mathrm{~m}$ from its base is $34.6^{\circ}$ . Find the height of the building.

Question 61

After leaving port, a ship sails $132 \mathrm{~km}$ in the direction $\mathrm{N} 47^{\circ} 49^{\prime} \mathrm{E}$ . Find how far north and how far east of the port the ship is now located.

Question 62

A rectangle measures $55 \mathrm{~cm}$ by $39 \mathrm{~cm}$ . Find the angles between the diagonals to the nearest degree.

Question 63

Calculate the area of a parallelogram:

Question 64

The cross-section of a canal will be a trapezoid $9.0 \mathrm{ft}$ . deep and $15 \mathrm{ft}$ . across the bottom, with walls sloping outward at an angle of $35^{\circ}$ . Calculate its cross-sectional area.

Question 65

The angle between the height and the slant height of a cone is $15.0^{\circ}$ . If the diameter of the circular base is $2.68 \mathrm{~cm}$ , what is the height of the cone?

Question 66

Draw this vector on coordinate axes: magnitude 7.5 , angle $16.1^{\circ}$

Question 67

Draw $262 / \underline{76.5^{\circ}}$ on coordinate axes.

Question 68

Determine the $x$ and $y$ components of a vector of magnitude 25 that makes a $19.2^{\circ}$ angle with the positive $x$ -axis.

Question 69

Determine the $x$ and $y$ components of a vector of magnitude 415 that makes a $69.2^{\circ}$ angle with the positive $x$ -axis.

Question 70

Determine the $x$ and $y$ components of a vector of magnitude 631 that makes a 22.15 rad angle with the positive $x$ -axis.

Question 71

Vector $\mathbf{A}$ (magnitude 0.374 ) and vector $\mathbf{B}$ (magnitude 0.604 $)$ are perpendicular. Find the magnitude of the resultant and the angle it makes with $\mathbf{B}$ .

Question 72

Vector A (magnitude 12) and vector $\mathbf{B}$ (magnitude 15) are perpendicular. Find the magnitude of the resultant and the angle it makes with $\mathbf{A}$ .

Question 73

Vectors $\mathbf{A}$ and $\mathbf{B}$ meet at right angles. Vector $\mathbf{A}$ has a magnitude of 678 and vector $\mathbf{B}$ has a magnitude of 876 . Calculate the angle the resultant makes with $\mathbf{B}$ .

Question 74

Find the resultant of $346 / \underline{147^{\circ}}$ and $563 / \underline{72.6^{\circ}}$ .

Question 75

Given the magnitude of vectors $\mathbf{A}$ and $\mathbf{B}$ , and the angle between them, find the magnitude of the resultant and the angle it makes with vector $\mathbf{B}: \mathbf{A}=529, \mathbf{B}=677, \theta=74.1^{\circ}$

Question 76

If $\mathbf{A}_{\mathbf{x}}=208$ and $\mathbf{A}_{\mathbf{y}}=378$ , find the magnitude of $\mathbf{A}$ and the angle $\theta$ it makes with the $x$ -axis.

Question 77

If $\mathbf{A}_{\mathbf{X}}=16.2$ and $\mathbf{A}_{\mathbf{y}}=4.1$ , find the magnitude of $\mathbf{A}$ and the angle $\theta$ it makes with the $x$ -axis.

Question 78

Combine these vectors to determine the resultant: $180 / \underline{7.4^{\circ}}$ and $216 / \underline{58.3^{\circ}}$

Question 79

If $\mathbf{V}_{\mathrm{x}}=2275$ and $\mathbf{V}$ makes an angle of $22.5^{\circ}$ with the $x$ -axis, find $\mathbf{V}_{\mathbf{y}}$ .

Question 80

If $\mathbf{V}_{\mathbf{y}}=17.5$ and $\mathbf{V}$ makes an angle of $42.0^{\circ}$ with the $x$ -axis, find $\mathbf{V}_{\mathbf{x}}$ .